= Cosmological soft-graviton consistency relation
{title2=$q\to0$}
A sufficiently long <adiabatic tensor mode> acts locally as an anisotropic change of spatial coordinates. Write the spatial metric as $a^2(e^\gamma)_{ij}$, with $\gamma_{ij}=\epsilon_{ij}^s\gamma^s$ transverse and trace-free. Removing a constant long mode maps the short <wavevector> to $\widetilde k_i=k_i-\gamma_{ij}k_j/2$ at first order. The trace-free deformation has no first-order volume Jacobian, so its effect on an isotropic scalar <power spectrum> is
$$
\delta P_\varphi(\mathbf k)=-\frac12\gamma_{ij}k_i\partial_{k_j}P_\varphi(k).
$$
Correlating this response with the long graviton gives
$$
\lim_{q\to0}\frac{\langle\gamma^s_{\mathbf q}\varphi_{\mathbf k}
\varphi_{-\mathbf k-\mathbf q}\rangle'}{P_\gamma(q)}
=-\frac12\epsilon_{ij}^s k_i\partial_{k_j}P_\varphi(k).
$$
The prime removes the momentum-conserving $(2\pi)^3\delta^3(\sum\mathbf k)$, and $P_\gamma$ is the power of the same polarization amplitude used in $\gamma_{ij}=\sum_s\epsilon_{ij}^s\gamma^s$. Thus polarization <normalization> must be consistent on both sides.<Non-Gaussian features of primordial fluctuations in single field inflationary models by Juan Maldacena>[Section 4.2, squeezed scalar-scalar-graviton limit]{ref}
This consistency relation concerns a <squeezed bispectrum configuration>: it relates a three-point <correlation function> to a coordinate response of a two-point function. It requires the long tensor to be adiabatic and the state/dynamics to respect that response. Additional physical long-wavelength effects or incompatible initial-state assumptions cannot simply be replaced by this identity.
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